AE502 Polarimetric Synthetic Aperture Radar

EE/Ge 157 b Week 2 Polarimetric Synthetic Aperture Radar (2)

POLARIMETRIC SYNTHETIC APERTURE RADAR COORDINATE SYSTEMS •

All matrices and vectors shown in this package are measured using the backscatter alignment coordinate system. This system is preferred when calculating radar-cross sections, and is used when measuring them:

Transmitting Antenna

hˆ t

vˆ t

zˆ

ˆ h r

kˆ t

kˆ r

  i

vˆ r Receiving Antenna

s

s

Scatterer

yˆ

i

xˆ

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POLARIMETRIC SYNTHETIC APERTURE RADAR MATHEMATICAL CHARACTERIZATION OF SCATTERERS: SCATTERING MATRIX

•

The radiated and scattered electric fields are related through the complex 2x2 scattering matrix:

E sc  Sprad •

The (complex) voltage measured at the antenna terminals is given by the scalar product of the receiving antenna polarization vector and the received wave electric field:

V  p rec Sp rad •

The measured power is the magnitude of the (complex) voltage squared:

P  VV *  p rec  Sprad

2

NOTE: Radar cross-section is proportional to power Jakob van Zyl

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POLARIMETRIC SYNTHETIC APERTURE RADAR MATHEMATICAL CHARACTERIZATION OF SCATTERERS: COVARIANCE MATRIX •

We can rewrite the expression for the voltage as follows: V  prec  Sprad rad rec rad  phrec phrad Shh  phrec pvrad Shv  prec v ph Svh  pv pv Svv

 phrec phrad

phrec pvrad

rad prec v ph

Shh  Shv  rad prec p v v  S   vh  Svv 

˜T A

•

The first vector contains only antenna parameters, while the second contains only scattering matrix elements. Using this expression in the power expression, one finds

  

P  VV *  AT TA  ATT*A*  A  C A* ; *

•

C  TT*

The matrix C is known as the covariance matrix of the scatterer

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POLARIMETRIC SYNTHETIC APERTURE RADAR MATHEMATICAL CHARACTERIZATION OF SCATTERERS: STOKES SCATTERING OPERATOR

•

The power expression can also be written in terms of the antenna Stokes vectors. First consider the following form of the power equation: P  prec  E sc prec E sc 

*

sc rec sc rec sc  phrec Ehsc  prec v Ev ph Eh  pv E v 

*

*  phrec phrec* Ehsc Ehsc *  pvrec pvrec* Evsc Evsc *   phrec pvrec* EhscEvsc*   pvrec prec EvscE hsc* h

phrec phrec*  Ehsc Ehsc*  pvrec pvrec*  Evsc Evsc*    rec rec*   sc sc*   phrec pvrec*  Ehsc Evsc*  pv ph  Ev Eh   grec  X

•

The vector X in the expression above is a function of the transmit antenna parameters as well as the scattering matrix elements.

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•

Using the fact that

E sc  Sprad ,

it can be shown that

X

can also be written as

X  Wgrad

where * * * Shh S*hh ShvShv ShhShv ShvShh  Svh S*vh Svv Svv* Svh Svv* Svv Svh*  W   * S S ShvSvv* Shh Svv* ShvSvh*   hh vh  * Svh Shh Svv S*hv Svh S*hv Svv S*hh 

•

This means that the measured power can also be expressed as: P  grec  W g red

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•

From the earlier definition of the Stokes vector, we note that the Stokes vector can be written as:  ph p*h  pv p*v  1 1 0 0ph p*h   ph p*h  pv p*v  1 1 0 0pv pv*  1 S      Rg  g  R S * *  *    0 0 1 1 p p  ph pv p p  h v  h v    * * i( ph pv  ph pv ) 0 0 i i p*h pv 

•

This means that we can express the measured power as:

~

P  Srec  R WR Srad  Srec  M Srad 1

•

1

The matrix M is known as the Stokes scattering operator. It is also called Stokes matrix.

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POLARIMETRIC SYNTHETIC APERTURE RADAR POLARIZATION SYNTHESIS

•

•

Once the scattering matrix, covariance matrix, or the Stokes matrix is known, one can synthesize the received power for any transmit and receive antenna polarizations using the polarization synthesis equations: Scattering matrix:

P  prec  Sprad

Covariance Matrix:

P  A CA*

Stokes scattering operator:

P  Srec  MSrad

2

Keep in mind that all matrices in the polarization synthesis equations must be expressed in the backscatter alignment coordinate system.

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POLARIMETRIC SYNTHETIC APERTURE RADAR

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POLARIMETRIC SYNTHETIC APERTURE RADAR Signatures

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POLARIMETRIC SYNTHETIC APERTURE RADAR POLARIMETER IMPLEMENTATION

•

To fully characterize the scatterer, one must measure the full scattering matrix: rec

 Eh    Shh Ev  Svh

•

tr

Shv Eh  Svv Ev 

Setting one of the elements of the transmit vector equal to zero allows one to measure two components of the scattering matrix at a time: S S S 1 inc S S S 0 inc   hh     hh hv   ;   hv     hh hv    Svh  Svh Svv 0 Svv  Svh Svv 1

•

This technique is commonly used to implement airborne and spaceborne SAR polarimeters, such as AIRSAR and SIR-C.

Jakob van Zyl

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POLARIMETRIC SYNTHETIC APERTURE RADAR POLARIMETER IMPLEMENTATION

TIMING

BLOCK DIAGRAM Horizontal

Transmission: Horizontal

Receiver Vertical

Transmitter Reception: Horizontal

Receiver

HH

Vertical

HH

VV

VH

HV

HH

VV

VH

Vertical VH

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HV

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POLARIMETRIC SYNTHETIC APERTURE RADAR POLARIZATION SIGNATURE OF DISTRIBUTED SCATTERERS •

• •

The previous examples showed polarization signatures of single scatterers. It can be shown that each of these examples exhibit up to two nulls in the copolarized signature. When forming multi-looked images, power is averaged to reduce speckle noise at the expense of spatial resolution. Mathematically, the multi-looking operation can be written as: N

N

i1

i 1

Pm   Pi   S

• • •

rec

 Mi S

rad

S

rec

 N  rad   Mi  S i1 

This means that the multi-looked polarization signature can be considered as the sum of the individual polarization signatures. The only way the multi-looked polarization signature can exhibit a null is if all the individual signatures that were added exhibited the same null. In general, the multi-looked signature will exhibit minima rather than nulls, and will appear to “sit on a pedestal.” The height of the pedestal is a measure of how different the individual polarization signatures are.

Jakob van Zyl

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POLARIMETRIC SYNTHETIC APERTURE RADAR Effect of Phase Error on Response of Trihedral Corner Reflector

No Error

45 Degrees

90 Degrees

180 Degrees Jakob van Zyl

135 Degrees 14

POLARIMETRIC SYNTHETIC APERTURE RADAR OBSERVED POLARIZATION SIGNATURES: SAN FRANCISCO

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POLARIMETRIC SYNTHETIC APERTURE RADAR OBSERVED POLARIZATION SIGNATURES: SAN FRANCISCO

At each frequency the ocean area scatters in a manner consistent with models of slightly rough surface scattering, the urban area like a dihedral corner reflector, while the park and natural terrain regions scatter much more diffusely, that is, the signatures possess large pedestals. This indicates the dominant scattering mechanisms responsible for the backscatter for each of the targets. The ocean scatter is predominantly single bounce, slightly rough surface scattering. The urban regions are characterized by two-bounce geometry as the incident waves are twice forward reflected from the face of a building to the ground and back to the radar, or vice versa. The apparent diffuse nature of the backscatter from the park and natural terrain indicates that in vegetated areas there exists considerable variation from pixel to pixel of the observed scattering properties, leading to the high pedestal. This variation may be due to multiple scatter or to a distinct variation in dominant scattering mechanism between 10m resolution elements.

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POLARIMETRIC SYNTHETIC APERTURE RADAR OBSERVED POLARIZATION SIGNATURES: L-BAND POLARIZATION SIGNATURES OF THE OCEAN

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POLARIMETRIC SYNTHETIC APERTURE RADAR OBSERVED POLARIZATION SIGNATURES: GEOLOGY

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POLARIMETRIC SYNTHETIC APERTURE RADAR OBSERVED POLARIZATION SIGNATURES: GEOLOGY

The previous viewgraph shows polarization signatures extracted from an AIRSAR image of the Pisgah lava flow in California. Signatures correspond to the flow itself (a very rough surface), an alluvial fan (medium roughness), and from the playa next to the flow (a very smooth surface). Note that as the roughness of the surface increases, so does the observed pedestal height. This is quite consistent with the predictions of the slightly rough surface models, even though the surface r.m.s. heights exceed the strict range of validity of the model. The exception is the playa case, where the pedestal at P-band is higher than that at L-band. Possible explanations for this behavior include subsurface scattering due to increased penetration at P-band, or signal-to-noise limitations for the very smooth surface.

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POLARIMETRIC SYNTHETIC APERTURE RADAR OBSERVED POLARIZATION SIGNATURES: SEA ICE

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POLARIMETRIC SYNTHETIC APERTURE RADAR OBSERVED POLARIZATION SIGNATURES: SEA ICE

Note the pedestal height as a function of frequency for the multi-year ice - the P-band pedestal is quite small, while the C-band pedestal is the greatest of the three. For the first year ice, exactly the opposite behavior is seen. The P-band signature shows the highest and the C-band signature the lowest pedestal. The multi-year ice behavior may be explained if we consider the ice to be formed of two layers, where the upper layer consists of randomly oriented oblong inclusions about the size of a C-band wavelength, several centimeters. The lower layer forms a solid, but slightly rough surface. In this situation the C-band signal would interact strongly with the diffuse scattering upper layer, giving rise to the high pedestal. The longer wavelength L- and P-band signals would pass through the upper and be scattered by the lower layer, which is smooth enough to exhibit fairly polarized backscatter. On the other hand, similar characteristics are also observed for simple rough surface scattering also, as previously explained. At present we have no model to explain the first year ice behavior.

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POLARIMETRIC SYNTHETIC APERTURE RADAR OBSERVED POLARIZATION SIGNATURES: VEGETATION

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POLARIMETRIC SYNTHETIC APERTURE RADAR OBSERVED POLARIZATION SIGNATURES: VEGETATION

The previous viewgraph polarization signatures for the heavily forested area and the clear-cut area of an area near Mt. Shasta in California. We note that all the signatures appear to be composed of a variable portion sitting on top of a large pedestal. The pedestal height varies with the type of scatterer, the heavy forest exhibiting a larger pedestal and the clear-cut a smaller pedestal. As discussed before, the pedestal is due to spatial variations in the observed scattering properties. From this we conclude that the returns from vegetated areas vary much more from pixel to pixels than the other terrain types discussed earlier. We also note that the pedestals increase with increasing wavelength due to increased penetration, plus the fact that the ratio of the size of the vegetation components relative to the wavelength is smaller for the longer wavelength.

Jakob van Zyl

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POLARIMETRIC SYNTHETIC APERTURE RADAR THEORETICAL CHARACTERISTICS: ODD NUMBERS OF REFLECTIONS

. x

Pixels dominated by odd numbers of reflections are typically characterized by Shh  Shv ;

Jakob van Zyl

Svv  Shv ;

* Shh Svv   Shv

2

* ; PhaseShh Svv   0

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POLARIMETRIC SYNTHETIC APERTURE RADAR THEORETICAL CHARACTERISTICS: EVEN NUMBERS OF REFLECTIONS

. .

x

Pixels dominated by odd numbers of reflections are typically characterized by Shh  Shv ;

Jakob van Zyl

Svv  Shv ;

* Shh Svv   Shv

2

* ; PhaseShh Svv   

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POLARIMETRIC SYNTHETIC APERTURE RADAR THEORETICAL CHARACTERISTICS: DIFFUSE SCATTERING

Pixels dominated by diffuse scattering are typically characterized by Shh  Shv ;

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Svv  Shv ;

* Shh Svv   Shv

2

* ; PhaseShh Svv   uniform0,2 

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POLARIMETRIC SYNTHETIC APERTURE RADAR UNSUPERVISED CLASSIFICATION OF SCATTERING MECHANISMS SINGLE FREQUENCY INTERPRETATION

MECHANISM

INTERPRETATION

Odd

Bare Surface, Heavy Vegetation

Even

Urban, Inundated Forest, Wetlands

Diffuse

“Moderate” Vegetation

The definition of “moderate” vegetation is a function of the frequency used when imaging the scene and can be shown to be related to the randomness of the orientation and the thickness of the scattering cylinders relative to the radar wavelength.

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POLARIMETRIC SYNTHETIC APERTURE RADAR Black Forest, Germany

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C-Band

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L-Band

P-Band

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C-Band

L-Band

P-Band

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C-Band

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L-Band

P-Band

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POLARIMETRIC SYNTHETIC APERTURE RADAR Landes Forest, France

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POLARIMETRIC SYNTHETIC APERTURE RADAR UNSUPERVISED CLASSIFICATION OF SCATTERING MECHANISMS

The average Stokes matrix of azimuthally symmetric terrain can be approximated by A  B0 1  B M   4  0  0

B A  B0 0 0

0 0 C  B0 D

0  0  D   B0  C

where 2

2

2

2

A  Shh  Svv B  Shh  Svv B0  2 Shv

2

* C  2 ReShh Svv 

D  2 ImShh Svv  *

In these expressions, the angular brackets denote ensemble averaging Jakob van Zyl

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C-Band

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L-Band

P-Band

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C-Band

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L-Band

P-Band

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POLARIMETRIC SYNTHETIC APERTURE RADAR CLOUDE’S DECOMPOSITION THEOREM

•

Cloude showed that a general covariance matrix  T can be decomposed as follows:  T  1k1 k1†   2k 2  k†2   3k3  k†3   4 k 4  k†4

•

•

Here, i ,i  1,2,3,4 are the eigenvalues of the covariance matrix, k i , i  1, 2,3, 4 are † its eigenvectors, and k i means the adjoint (complex conjugate transposed) of .k i In the monostatic (backscatter) case, the covariance matrix has one zero eigenvalue, and the decomposition results in at most three nonzero covariance matrices.

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•

Also useful in our discussions later is Cloude's definition of target entropy, HT 

•

4

 Pi log 4 Pi 

i1

where  Pi  4 i

 j j1

•

As pointed out by Cloude, the target entropy is a measure of target disorder, with HT  1 for random targets and HT  0 for simple (single) targets.

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POLARIMETRIC SYNTHETIC APERTURE RADAR CLOUDE’S DECOMPOSITION THEOREM AZIMUTHALLY SYMMETRIC NATURAL TERRAIN

•

Borgeaud et al. showed, using a random medium model, that the average covariance matrix for azimuthally symmetrical terrain in the monostatic case has the general form  1 0   T  C  0  0      * 0  

•

where C  Shh S*hh

•



* Shh Svv

Shh S*hh



* 2 Shv Shv

Shh S*hh



* Svv Svv

* Shh Shh

The superscript * means complex conjugate, and all quantities are ensemble averages. The parameters C, ,  and  all depend on the size, shape and electrical properties of the scatterers, as well as their statistical angular distribution.

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•

The eigenvalues of  T are 1 

C   1  2 

 12  4  2 

2 

C   1  2 

 12  4  2 

 3  C

•

Note that the three eigenvalues are always real numbers greater than or equal to zero.

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•

The corresponding three eigenvectors are

k1 

k2 

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2     1    2  1     0 2    1    4  2   1    









2     1    2  1    0 2    1    4  2   1    









0  k 3   1      0 

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•

On the previous page we used the shorthand notation 2

   1  4  2

•

We note that  is always positive. Also note that we can write



 1   k11  2 k21 4

•

where K

•



2

K

  1   2  1    1   2  1 

2  4  2  2    4  2   

Since K is always positive, it follows that the ratio of k11 to k 21 is always negative. This means that the first two eigenvectors represent scattering matrices that can be interpreted in terms of odd and even numbers of reflections.

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POLARIMETRIC SYNTHETIC APERTURE RADAR Example of Eigenvalue Decomposition San Francisco, California

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POLARIMETRIC SYNTHETIC APERTURE RADAR CLOUDE'S DECOMPOSITION THEOREM RANDOMLY ORIENTED DIELECTRIC CYLINDERS

•

In general, the scattering matrix of a single dielectric cylinder oriented horizontally can be written as a 0

S    • •

0  b

where a and b are complex numbers whose magnitudes and phases are functions of cylinder dielectric constant, diameter and length. Assuming a uniform distribution in angles about the line of sight, one can easily show that the resulting average covariance matrix for the monostatic case has the following parameters C



 

1 3 a 2  3 b 2  2 a *b 8

   3 a 2  3 b 2  2 a *b  a 2  b 2  6 a *b

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 1



2 a  b2

 

3 a 2  3 b 2  2 a*b

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•

The eigenvalues are: 1  C1  

•

 2  C1   

 3  C

The corresponding three eigenvectors are    1   k1  0   2    1 

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    1   k2  0   2    1 

0  k 3   1      0 

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•

In the thin cylinder limit, b  0 , and we find that thin 1 / 3

thin  2 / 3  1 0 2   T  C  0 0 3     1 0

•

• •

1  1 0 1 0 1  2    0   0 0 0   0  3   3  1 1 0 1  0

0 0  1 0   0 0 

In this case, equal amounts of scattering is contributed by the matrix that resembles scattering by a sphere and by the cross-polarized return, although a significant fraction of the total energy is also contained in the second matrix, which resembles a metal dihedral corner reflector. The entropy in this case is 0.95 indicating a high degree of target disorder or randomness. Note that the unsupervised classification scheme would classify this as diffuse scattering.

Jakob van Zyl

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•

In the thick cylinder limit, b  a and we find that thick  1

thick  0

• • •

•

In this case, only one eigenvalue is non-zero, and the average covariance matrix is identical to that of a sphere. The entropy is 0, indicating no target randomness, even though we have calculated the average covariance matrix for randomly oriented thick cylinders! The explanation for this result lies in the fact that when the cylinders are thick, the single cylinder scattering matrix becomes the identity matrix, which is insensitive to rotations. Note that the unsupervised classification scheme would classify this as odd numbers of reflections.

Jakob van Zyl

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POLARIMETRIC SYNTHETIC APERTURE RADAR CLOUDE'S DECOMPOSITION THEOREM RADAR THIN VEGETATION INDEX

•

Using the result for a cloud of randomly oriented thin cylinders, we note that 3 1  1  2  3 4

•

We now define a radar thin vegetation index (RVI) as * * 8 ShvShv ShvShv 4 3 4 l RVI    ; l  * 1  2  3 ShvShh  Svv Svv*  2 ShvS*hv 1  l Svv Svv*

•

We expect RVI to vary between 0 and 1.

Jakob van Zyl

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POLARIMETRIC SYNTHETIC APERTURE RADAR CLOUDE'S DECOMPOSITION THEOREM RADAR THIN VEGETATION INDEX

1 0.8 0.6

RVI

a 0 0 b

S  

0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

Ratio of b to a Ratio of b to a

Jakob van Zyl

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POLARIMETRIC SYNTHETIC APERTURE RADAR CLOUDE'S DECOMPOSITION THEOREM: EXAMPLE VEGETATED CLEARCUT AREA

Parameter

P-Band

L-Band

C-Band



0.5621

0.5308

0.4083



0.5642

0.7580

0.7159



0.0928+i0.0582

0.2324+i0.1057

0.3558+i0.0440

1

1.0260

1.1615

1.2437

2

0.5382

0.5964

0.4722

3

0.5261

0.5308

0.4083

Entropy

0.95

0.94

0.88

RVI

1.01

0.93

0.77

Jakob van Zyl

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POLARIMETRIC SYNTHETIC APERTURE RADAR CLOUDE'S DECOMPOSITION THEOREM: EXAMPLE

The clearcut area is covered with short shrub-like vegetation. We note that at all three frequencies the scattering is dominated by an odd number of reflections, i.e the first eigenvalue is the dominant one. At P-band the even number of reflections (second eigenvalue) and the cross-polarized returns (third eigenvalue) are almost the same strength, and about half that of the odd numbers of reflections. This is very similar to the thin randomly oriented cylinder case discussed earlier. As the frequency increases, the even number of reflections and the cross-polarized returns become more different, and also become a smaller fraction of the total scattering. This is consistent with the randomly oriented cylinder case where the radius of the cylinder increases. The same conclusion is reached when considering the entropy. The highest value (most randomness) is observed at P-band, and the randomness decreases with increasing frequency. As pointed out before, as the cylinder radius increases, the entropy decreases. Therefore, from the decomposition results we conclude that the vegetation in the clearcut is randomly oriented, and that most of the scattering comes from vegetation that have branches that are thin compared to the three radar wavelengths.

Jakob van Zyl

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C-Band

Jakob van Zyl

L-Band

P-Band

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C-Band

Jakob van Zyl

L-Band

P-Band

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•

Advantages – –

•

Rigorous mathematical technique Provides quantitative information about scattering mechanisms

Disadvantages – –

Jakob van Zyl

Not physically based Interpretation of results not unique

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